In calculus, Rolle's theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary or critical point somewhere between them—that is, a point where the first derivative (the slope of the tangent line) is equal to zero. Use these definition of Rolles Theorem to determine the Rolles Theorem of the following functions. 

Example 

Verify Rolles Theorem for the following functions.
(i) ƒ(x)=x³-4x²+3x+3 in [0,3] 

(ii) SOLUTION : 

(i) To verify Rolles Theorem for ƒ(x)=x³-4x²+3x+3 in [0,3] is pretty simple  since the function is a polynomial function that is continuous on [0,3] and differentiable on (0,3), substitute the given interval in the function to verify Rolles Theorem. Then f(0)=3 and f(3)=3³-4(3)²+3(3)+3=3 hence f(0)=f(3)=3 which satisfies Rolles Theorem, next we take the derivative of f(x) which yields f'(x)=3x²-8x+3 now take the stationary point f'(c)=0.

f'(c)=3c²-8c+3=0 

Hence c=2.215 and c=0.451 which are the stationary points of the function. Although stationary points is not an element of the interval (0,1),still the function satisfies Rolles Theorem. 

(ii) the function is is a composite function Involving a polynomial and an exponential function so that makes it a lil bit complex, now let's check if Rolles Theorem is Applicable. 


The function is continuous on [0,1] and differentiable on(0,1). Substitute the interval in the function I.e f(0)=0 and f(1)=0, 
f(0)=f(1)=0 which satisfies the Rolles Theorem. Stationary point for this function does not exist but the function still satisfies Rolles Theorem.